Maps, Hash Tables and Dictionaries

Chapter 10.1, 10.2, 10.3, 10.5

Outline

Maps

Hashing

Dictionaries

Ordered Maps & Dictionaries

Outline

Maps

Hashing

Dictionaries

Ordered Maps & Dictionaries

Maps

A map models a searchable collection of key-value entries

The main operations of a map are for searching, inserting, and deleting items

Multiple entries with the same key are not allowed

Applications: address book

student-record database

The Map ADT

Map ADT methods: get(k): if the map M has an entry with key k, return its associated

value; else, return null

put(k, v): insert entry (k, v) into the map M; if key k is not already in M, then return null; else, return old value associated with k

remove(k): if the map M has an entry with key k, remove it from M and return its associated value; else, return null

size(), isEmpty()

keys(): return an iterator over the keys in M

values(): return an iterator of the values in M

entries(): returns an iterator over the entries in M

ExampleOperation Output M

isEmpty() true Ø

put(5,A) null (5,A)

put(7,B) null (5,A),(7,B)

put(2,C) null (5,A),(7,B),(2,C)

put(8,D) null (5,A),(7,B),(2,C),(8,D)

put(2,E) C (5,A),(7,B),(2,E),(8,D)

get(7) B (5,A),(7,B),(2,E),(8,D)

get(4) null (5,A),(7,B),(2,E),(8,D)

get(2) E (5,A),(7,B),(2,E),(8,D)

size() 4 (5,A),(7,B),(2,E),(8,D)

remove(5) A (7,B),(2,E),(8,D)

remove(2) E (7,B),(8,D)

get(2) null (7,B),(8,D)

isEmpty() false (7,B),(8,D)

Comparison with java.util.Map

Map ADT Methods java.util.Map Methods

size() size()

isEmpty() isEmpty()

get(k) get(k)

put(k,v) put(k,v)

remove(k) remove(k)

keys() keySet()

values() values()

entries() entrySet()

A Simple List-Based Map

We could implement a map using an unsorted list We store the entries of the map in a doubly-linked list S, in

arbitrary order

S supports the node list ADT (Section 6.2)

trailerheader nodes/positions

entries

9 c 6 b 5 a 8 d

The get(k) Algorithm

Algorithm get(k):

B = S.positions() {B is an iterator of the positions in S}

while B.hasNext() do

p = B.next() // the next position in B

if p.element().getKey() = k then

return p.element().getValue()

return null {there is no entry with key equal to k}

The put(k,v) Algorithm

Algorithm put(k,v):

B = S.positions()

while B.hasNext() do

p = B.next()

if p.element().getKey() = k then

t = p.element().getValue()

S.set(p,(k,v))

return t {return the old value}

S.addLast((k,v))

n = n + 1 {increment variable storing number of entries}

return null {there was no previous entry with key equal to k}

The remove(k) Algorithm

Algorithm remove(k):

B =S.positions()

while B.hasNext() do

p = B.next()

if p.element().getKey() = k then

t = p.element().getValue()

S.remove(p)

n = n – 1 {decrement number of entries}

return t {return the removed value}

return null {there is no entry with key equal to k}

Performance of a List-Based Map

Performance: put, get and remove take O(n) time since in the worst case

(the item is not found) we traverse the entire sequence to look for an item with the given key

The unsorted list implementation is effective only for small maps

Outline

Maps

Hashing

Dictionaries

Ordered Maps & Dictionaries

Hash Tables

A hash table is a data structure that can be used to make map operations faster.

While worst-case is still O(n), average case is typically O(1).

Applications of Hash Tables

databases

compilers

browser caches

Hash Functions and Hash Tables A hash function h maps keys of a given type to integers

in a fixed interval [0, N - 1]

Example:h(x) = x mod N

is a hash function for integer keys

The integer h(x) is called the hash value of key x

A hash table for a given key type consists of Hash function h

Array (called table) of size N

When implementing a map with a hash table, the goal is to store item (k, o) at index i = h(k)

Example

We design a hash table for a map storing entries as (SIN, Name), where SIN (social insurance number) is a nine-digit positive integer

Our hash table uses an array of size N = 10,000 and the hash functionh(x) = last four digits of SIN x

Ø

Ø

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Ø

01234

999799989999

…

451-229-0004

981-101-0002

200-751-9998

025-612-0001

Hash Functions

A hash function is usually specified as the composition of two functions:

Hash code: h1: keys integers

Compression function: h2: integers [0, N - 1]

The hash code is applied first, and the compression function is applied next on the result, i.e.,

h(x) = h2(h1(x))

The goal of the hash function is to “disperse” the keys in an apparently random way

Hash Codes Memory address:

We reinterpret the memory address of the key object as an integer (default hash code of all Java objects)

Does not work well when copies of the same object may be stored at different locations.

Integer cast: We reinterpret the bits of the key as an integer

Suitable for keys of length less than or equal to the number of bits of the integer type (e.g., byte, short, int and float in Java)

Component sum: We partition the bits of the key into components of fixed length (e.g.,

16 or 32 bits) and we sum the components (ignoring overflows)

Suitable for keys of fixed length greater than or equal to the number of bits of the integer type (e.g., long and double in Java)

Problems with Component Sum Hash Codes

Hashing works when the number of different common keys is small relative to the

hashing space (e.g., 232 for a 32-bit hash code).

the hash codes for common keys are well-distributed (do not collide) in this space.

Component Sum codes ignore the ordering of the components. e.g., using 8-bit ASCII components, ‘stop’ and ‘pots’ yields the

same code.

Since common keys are often anagrams of each other, this is often a bad idea!

Polynomial Hash Codes Polynomial accumulation:

We partition the bits of the key into a sequence of components of fixed length (e.g., 8, 16 or 32 bits) a0 a1 … an-1

We evaluate the polynomial

p(z) = a0 + a1 z + a2 z2 + … + an-1zn-1 at a fixed value z, ignoring overflows

Especially suitable for strings Polynomial p(z) can be evaluated in O(n) time using Horner’s rule:

The following polynomials are successively computed, each from the previous one in O(1) time

p0(z) = an-1

pi (z) = an-i-1 + zpi-1(z) (i = 1, 2, …, n -1)

We have p(z) = pn-1(z)

Compression Functions

Division: h2 (y) = y mod N

The size N of the hash table is usually chosen to be a prime (on the assumption that the differences between hash keys y are less likely to be multiples of primes).

Multiply, Add and Divide (MAD): h2 (y) = [(ay + b) mod p] mod N, where

p is a prime number greater than N

a and b are integers chosen at random from the interval [0, p – 1], with a > 0.

Collision Handling

Collisions occur when different elements are mapped to the same cell

Separate Chaining: Let each cell in the table point to a linked list of entries that map

there

Separate chaining is simple, but requires additional memory outside the table

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01234 451-229-0004 981-101-0004

025-612-0001

Map Methods with Separate Chaining

Delegate operations to a list-based map at each cell:

Algorithm get(k):

Output: The value associated with the key k in the map, or null if there is no entry with key equal to k in the map

return A[h(k)].get(k) {delegate the get to the list-based map at A[h(k)]}

Map Methods with Separate Chaining

Delegate operations to a list-based map at each cell:

Algorithm put(k,v):

Output: Store the new (key, value) pair. If there is an existing entry with key equal to k, return the old value; otherwise, return null

t = A[h(k)].put(k,v) {delegate the put to the list-based map at A[h(k)]}

if t = null then {k is a new key}n = n + 1

return t

Map Methods with Separate Chaining

Delegate operations to a list-based map at each cell:

Algorithm remove(k):

Output: The (removed) value associated with key k in the map, or null if there

is no entry with key equal to k in the map

t = A[h(k)].remove(k) {delegate the remove to the list-based map at A[h(k)]}

if t ≠ null then {k was found}

n = n - 1

return t

Open Addressing: Linear Probing

Open addressing: the colliding item is placed in a different cell of the table

Linear probing handles collisions by placing the colliding item in the next (circularly) available table cell

Each table cell inspected is referred to as a “probe”

Colliding items lump together, so that future collisions cause a longer sequence of probes

Example:

h(x) = x mod 13

Insert keys 18, 41, 22, 44, 59, 32, 31, 73, in this order

0 1 2 3 4 5 6 7 8 9 10 11 12 41 18 44 59 32 22 31 73

Get with Linear Probing

Consider a hash table A of length N that uses linear probing

get(k) We start at cell h(k)

We probe consecutive locations until one of the following occurs

An item with key k is found, or

An empty cell is found, or

N cells have been unsuccessfully probed

Algorithm get(k)i h(k)p 0repeat

c A[i]if c = Ø

return null

else if c.key () = kreturn

c.element()else

i (i + 1) mod N

p p + 1until p = N

return null

Remove with Linear Probing

Suppose we receive a remove(44) message.

What problem arises if we simply remove the key = 44 entry?

Example:

h(x) = x mod 13

Insert keys 18, 41, 22, 44, 59, 32, 31, 73, in this order

0 1 2 3 4 5 6 7 8 9 10 11 12 41 18 44 59 32 22 31 73

k h(k) i

18 5 5

41 2 2

22 9 9

44 5 6

59 7 7

32 6 8

31 5 10

73 8 11

✗Ø

Removal with Linear Probing To address this problem, we introduce a special object, called

AVAILABLE , which replaces deleted elements

AVAILABLE has a null key

No changes to get(k) are required.

Algorithm get(k)i h(k)p 0repeat

c A[i]if c = Ø

return null

else if c.key () = kreturn

c.element()else

i (i + 1) mod N

p p + 1until p = N

return null

Updates with Linear Probing

remove(k)

We search for an entry with key k

If such an entry (k, o) is found, we replace it with the special item AVAILABLE and we return element o

Else, we return null

put(k, o)

We throw an exception if the table is full

We start at cell h(k)

We probe consecutive cells until one of the following occurs A cell i is found that is either empty or stores AVAILABLE, or

N cells have been unsuccessfully probed

We store entry (k, o) in cell i

Open Addressing: Double Hashing Double hashing is an alternative open addressing method that uses

a secondary hash function h’(k) in addition to the primary hash function h(x).

Suppose that the primary hashing i=h(k) leads to a collision.

We then iteratively probe the locations(i + jh’(k)) mod N for j = 0, 1, … , N - 1

The secondary hash function h’(k) cannot have zero values

N is typically chosen to be prime.

Common choice of secondary hash function h’(k): h’(k) = q - k mod q, where

q < N

q is a prime

The possible values for h’(k) are 1, 2, … , q

Consider a hash table storing integer keys that handles collision with double hashing N = 13

h(k) = k mod 13

h’(k) = 7 - k mod 7

Insert keys 18, 41, 22, 44, 59, 32, 31, 73

Example of Double Hashing

0 1 2 3 4 5 6 7 8 9 10 11 12

31 41 18 32 59 73 22 44

Consider a hash table storing integer keys that handles collision with double hashing N = 13

h(k) = k mod 13

h’(k) = 7 - k mod 7

Insert keys 18, 41, 22, 44, 59, 32, 31, 73

Example of Double Hashing

0 1 2 3 4 5 6 7 8 9 10 11 12

31 41 18 32 59 73 22 44

Performance of Hashing In the worst case, searches, insertions and removals on a hash table

take O(n) time

The worst case occurs when all the keys inserted into the map collide

The load factor λ = n/N affects the performance of a hash table For separate chaining, performance is typically good for λ < 0.9.

For open addressing , performance is typically good for λ < 0.5.

java.util.HashMap maintains λ < 0.75

Open addressing can be more memory efficient than separate chaining, as we do not require a separate data structure.

However, separate chaining is typically as fast or faster than open addressing.

Rehashing

When the load factor λ exceeds threshold, the table must be rehashed. A larger table is allocated (typically at least double the size).

A new hash function is defined.

All existing entries are copied to this new table using the new hash function.

Outline

Maps

Hashing

Dictionaries

Dictionary ADT The dictionary ADT models a

searchable collection of key-element entries

The main operations of a dictionary are searching, inserting, and deleting items

Multiple items with the same key are allowed

Applications: word-definition pairs

credit card authorizations

Dictionary ADT methods:

get(k): if the dictionary has at least one entry with key k, returns one of them, else, returns null

getAll(k): returns an iterable collection of all entries with key k

put(k, v): inserts and returns the entry (k, v)

remove(e): removes and returns the entry e. Throws an exception if the entry is not in the dictionary.

entrySet(): returns an iterable collection of the entries in the dictionary

size(), isEmpty()

Dictionaries and Java

Note: The java.util.Dictionary class actually implements a map ADT.

There is no dictionary data structure in the Java Collections Framework that supports multiple entries with equal keys.

The textbook (Section 9.5.3) provides an implementation of a Dictionary based upon a map of keys, each entry of which supports a linked list of entries with the same key.

ExampleOperation Output Dictionary

put(5,A) (5,A) (5,A)

put(7,B) (7,B) (5,A),(7,B)

put(2,C) (2,C) (5,A),(7,B),(2,C)

put(8,D) (8,D) (5,A),(7,B),(2,C),(8,D)

put(2,E) (2,E) (5,A),(7,B),(2,C),(8,D),(2,E)

get(7) (7,B) (5,A),(7,B),(2,C),(8,D),(2,E)

get(4) null (5,A),(7,B),(2,C),(8,D),(2,E)

get(2) (2,C) (5,A),(7,B),(2,C),(8,D),(2,E)

getAll(2) (2,C),(2,E) (5,A),(7,B),(2,C),(8,D),(2,E)

size() 5 (5,A),(7,B),(2,C),(8,D),(2,E)

remove(get(5)) (5,A) (7,B),(2,C),(8,D),(2,E)

get(5) null (7,B),(2,C),(8,D),(2,E)

Subtleties of remove(e) remove(e) will remove an entry that matches e (i.e., has

the same (key, value) pair).

If the dictionary contains more than one entry with identical (key, value) pairs, remove(e) will only remove one.

Example:Operation Output Dictionary

e1 = put(2,C) (2,C) (5,A),(7,B),(2,C)

e2 = put(8,D) (8,D) (5,A),(7,B),(2,C),(8,D)

e3 = put(2,E) (2,E) (5,A),(7,B),(2,C),(8,D),(2,E)

remove(get(5)) (5,A) (7,B),(2,C),(8,D),(2,E)

remove(e3) (2,E) (7,B),(2,C),(8,D)

remove(e1) (2,C) (7,B),(8,D)

A List-Based Dictionary

A log file or audit trail is a dictionary implemented by means of an unsorted sequence

We store the items of the dictionary in a sequence (based on a doubly-linked list or array), in arbitrary order

Performance:

insert takes O(1) time since we can insert the new item at the beginning or at the end of the sequence

find and remove take O(n) time since in the worst case (the item is not found) we traverse the entire sequence to look for an item with the given key

The log file is effective only for dictionaries of small size or for dictionaries on which insertions are the most common operations, while searches and removals are rarely performed (e.g., historical record of logins to a workstation)

The getAll and put Algorithms

Algorithm getAll(k)

Create an initially-empty list L

for e: D do

if e.getKey() = k then

L.addLast(e)

return L

Algorithm put(k,v)

Create a new entry e = (k,v)

S.addLast(e) {S is

unordered}

return e

The remove Algorithm

Algorithm remove(e):

{ We don’t assume here that e stores its position in S }

B = S.positions()

while B.hasNext() do

p = B.next()

if p.element() = e then

S.remove(p)

return e

return null {there is no entry e in D}

Hash Table Implementation

We can also create a hash-table dictionary implementation.

If we use separate chaining to handle collisions, then each operation can be delegated to a list-based dictionary stored at each hash table cell.

Outline

Maps

Hashing

Dictionaries

Ordered Maps & Dictionaries

Ordered Maps and Dictionaries If keys obey a total order relation, can represent a map or

dictionary as an ordered search table stored in an array.

Can then support a fast find(k) using binary search.

at each step, the number of candidate items is halved

terminates after a logarithmic number of steps

Example: find(7)

1 3 4 5 7 8 9 11 14 16 18 19

1 3 4 5 7 8 9 11 14 16 18 19

1 3 4 5 7 8 9 11 14 16 18 19

1 3 4 5 7 8 9 11 14 16 18 19

0

0

0

0

ml h

ml h

ml h

l=m =h

Ordered Search Tables Performance:

find takes O(log n) time, using binary search

insert takes O(n) time since in the worst case we have to shift n items to make room for the new item

remove takes O(n) time since in the worst case we have to shift n items to compact the items after the removal

A search table is effective only for dictionaries of small size or for dictionaries on which searches are the most common operations, while insertions and removals are rarely performed (e.g., credit card authorizations)

Summary: Learning Outcomes Maps

ADT

What are they good for?

Naïve implementation – running times

Hashing Running time

Types of hashing

Dictionaries ADT

What are they good for?

Ordered Maps & Dictionaries What are they good for?

Running time